Document 6574189

International Journal of Partial Differential Equations and Applications, 2014, Vol. 2, No. 4, 79-85
Available online at http://pubs.sciepub.com/ijpdea/2/4/4
© Science and Education Publishing
DOI:10.12691/ijpdea-2-4-4
Analytical investigation of the Laminar Viscous Flow in
a Semi-Porous Channel in the Presence of a Uniform
Magnetic Field
A. Majidian, M. Fakour*, A. Vahabzadeh
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
*Corresponding author: [email protected], [email protected]
Received October 09, 2014; Revised October 20, 2014; Accepted October 24, 2014
Abstract In this paper, the laminar fluid flow in a semi-porous channel in the presence of transverse magnetic
field is investigated. The homotopy perturbation method (HPM) is employed to compute an approximation to the
solution of the system of nonlinear differential equations governing the problem. It has been attempted to exhibit the
reliability and performance of the homotopy perturbation method (HPM) in comparison with the numerical method
(Richardson extrapolation) in solving this problem. The influence of the two dimensionless numbers: the Hartmann
number and Reynolds number on non-dimensional velocity profile are considered. The results indicate that velocity
boundary layer thickness decrease with increase of Reynolds number and it increases as Hartmann number increases.
Keywords: homotopy perturbation method (HPM), laminar viscous flow, Semi-porous channel, uniform magnetic
field
Cite This Article: A. Majidian, M. Fakour, and A. Vahabzadeh, “Analytical investigation of the Laminar
Viscous Flow in a Semi-Porous Channel in the Presence of a Uniform Magnetic Field.” International Journal of
Partial Differential Equations and Applications, vol. 2, no. 4 (2014): 79-85. doi: 10.12691/ijpdea-2-4-4.
1. Introduction
The flow problem in porous tubes or channels received
much attention in recent years because of its various
applications in biomedical engineering, for example, in
the dialysis of blood in artificial kidney [1], in the flow of
blood in the capillaries [2], in the flow in blood
oxygenators [3], as well as in many other engineering
areas such as the design of filters [4], in transpiration
cooling boundary layer control [5] and gaseous diffusion
[6]. In 1953, Berman [7] described an exact solution of the
Navier-Stokes equation for steady two dimensional
laminar flow of a viscous, incompressible fluid in a
channel with parallel, rigid, porous walls driven by
uniform, steady suction or injection at the walls. This
mass transfer is paramount in some industrial processes.
In the pastyears, many studies have been published on
problems related to the influence of an induction on the
dynamic and thermodynamic fields, Osterle and Young [8]
and Umavathi [9] considered the natural convection
between two parallel plates, Askovic et al. [10] or Askovic
[11] presented some developments of three dimensional
MHD problems in case of non-stationary flows or flows
induced by accelerated or deformable bodies. More
recently, Chandran and Sacheti [12] analyzed the effects
of a magnetic field on the thermodynamic flow past a
continuously moving porous plate.
Nomenclature
HPM: Homotopy perturbation method
NUM: Numerical method
P: Pressure
q: Mass transfer parameter
Re: Reynolds number
U: Dimensionless velocity in the x direction
V: Dimensionless velocity in the y direction
h: Suspension height
Ha: Hartman number
Lx: Length of the slider
u0: x velocity of the pad
u: Dimensionless x −component velocity
v: Dimensionless y −component velocity
u*: Velocity component in the x direction
v*: Velocity component in the y direction
x: Dimensionless horizontal coordinate
y: Dimensionless vertical coordinate
x*: Distance in the x direction parallel to the plates
y*: Distance in the y direction parallel to the plates
Greek Symbols
ρ : Fluid density
ν : Kinematic viscosity
σ : Electrical conductivity
ε : Aspect ratio h / Lx
Simultaneously, Chamkha [13,14] has detailed the
influence of a magnetic field on an electrically conducting
fluid in the neighborhood of a cone, a wedge or near a
stagnation point. More recently, Desseaux [15] analyzed
the influence of a magnetic field over a laminar viscous
flow in a semi-porous channel. All these problems and
phenomena are modeled by ordinary or partial differential
International Journal of Partial Differential Equations and Applications
equations. d In recent years, much attention has been
devoted to the newly developed methods to construct an
analytical solution of equation. In this paper, the basic
idea of the HPM is introduced and then the coupled
ordinary non-linear differential equations of the laminar
viscous flow in a semi-porous channel are solved through
the HPM. The effects of the Hartmann number and
Reynolds number on velocity profile are considered.
magnetic field B is assumed to be applied towards
direction y* (Figure 1) [15].
In the case of a short circuit to neglect the electrical
field and perturbations to the basic normal field and
without any gravity forces, according to Osterle and
Young [8], Umavathi [9], the governing equations are
∂u* ∂v*
+
=
0,
∂x* ∂y*
2. Describe Problem and Mathematical
Formulation
Consider the laminar two-dimensional stationary flow
of an electrically conducting incompressible viscous fluid
in a semi-porous channel made by a long rectangular plate
of length Lx in uniform translation in x* direction and an
infinite porous plate. The distance between the two plates
is h. The physical fluid properties ( ρ , µ ) are constant. We
observe a normal velocity q at the porous wall. A uniform
80
u*
(1)
∂u* * ∂u*
+v
∂x*
∂y*
2
 ∂ 2 u * ∂ 2 u* 
1 ∂P*
 − u* σ B ,
=
−
+ν 
+
 ∂x*2 ∂y*2 
ρ ∂x*
ρ


u*
(2)
 ∂ 2v* ∂ 2v* 
∂v* * ∂v*
1 ∂P*
 . (3)
+v
=
−
+ν 
+
 ∂x*2 ∂y*2 
ρ ∂y*
∂x*
∂y*


Figure 1. Schematic of the problem (fluid in a porous media between parallel plates and magnetic field)
The appropriate boundary conditions for the velocity
are
=
y*
0=
: u*
u0*=
, v*
0,
=
y* h=
: u* 0,=
v* 0.
(4)
h
∫u
* × dy* =
Lx × q.
Re =
(5)
We consider the following transformations
Lx
u
=
h
u*
v*
P*
=
=
,v
, Py
.
U
q
ρ .q 2
ν
(9)
(10)
.
∂u ∂v
+ =
0,
∂x ∂y
(6)
0
*
*
x= x , y= y ,
hq
σ
,
ρ .ν
Introducing Eqs. (6) and (10) in Eqs. (1) and (3) leads
to the dimensionless equations
Calculating a mean velocity U by the relation
U ×h =
Ha = Bh
u
(7)
∂u
∂u
+v
∂x
∂y
∂Py ν  ∂ 2u ∂ 2u 
Ha 2
=
−ε
+ ε 2 + 2  − u
,
∂x hq  ∂x
Re
∂y 
(12)
2

(8)
Then, we can consider two dimensionless numbers: the
Hartman number Ha for the description of the magnetic
forces [15] and the Reynolds number Re for the dynamic
forces
(11)
u

∂P ν  ∂ 2v ∂ 2v 
∂v
∂v
+v =
− y +  ε 2 2 + 2  . (13)
∂x
∂y
∂y hq  ∂x ∂y 


Quantity ε is defined as the ratio between distance h
and a characteristic length Lx of the slider. This ratio is
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International Journal of Partial Differential Equations and Applications
normally small. Berman’s similarity transformation is
used to be free from the aspect ratio ε:
u*
U
u0 .U ( y) + x
v=
−V ( y ) ,=
u=
dV
.
dy
(14)
Ha 2
1
1 ∂Py
ε2
V ′′′ +
V′ =
,
x ∂x
Re
Re
(15)
1 
U ′′ − Ha 2U  .

Re 
(16)
UV ′ − VU=′
The right-hand side of Eq. (15) is constant. So, we
derive this equation with respect to y. This gives
=
V
Ha V ′′ + Re [V ′V ′′ − VV ′′′] .
IV
2
(17)
where primes denote differentiation with respect to y and
asterisks have been omitted for simplicity. The dynamic
boundary conditions become
=
y 0 :=
U 1,=
V 0,=
V ′ 0,
=
y 1:=
U 0,=
V 1,=
V ′ 0.
(18)
(19)
3. Describe Homotopy Perturbation
Method and Applied to the Problem
3.1. Describe Homotopy Perturbation Method
Consider the following function:
A(u ) − f (r ) =
0
(20)
∂u
)=0
∂n
(21)
where A (u) is defined as follows:
A=
(u ) L(u ) + N (u )
(22)
Homotopy-perturbation structure is shown as:
H (ν =
, p) L(ν ) − L(u0 ) + p.L(u0 )
0
+ p [ N (ν ) − f (r ) ] =
u = limν = ν 0 + ν 1 + ν 2 + 
the above convergence is discussed in [16,19].
3.2. The HPM Applied to the Problem
A homotopy perturbation method can be constructed as
follows:
H (U , p) =(1 − p)(−
1
U ′′) +
Re
 V IV − Ha 2 V ′′

(1 − p )( V IV ) + p 
H ( V , p) =
 . (30)
 + Re ( VV ′′′ − V ′V ′′ ) 


One can now try to obtain a solution of Eqs. (29, 30) in
the form of:
U ( y ) =U 0 ( y ) + pU1 ( y ) + p 2U 2 ( y ) +
(31)
V ( y ) =V0 ( y ) + pV1 ( y ) + p 2V2 ( y ) +
(32)
where vi(Y), i=1,2,3,… are functions yet to be determined.
According to Eqs. (29,30) the initial approximation to
satisfy boundary condition is:
U 0 ( y ) =− y + 1,
(33)
V0 ( y ) =
−2 y 3 + 3 y 2 .
(34)
Substituting Eqs.(31, 32) into Eqs. (29, 30) yields:
1 − 1 y + 0.27 y 4 − 0.021 y8 + 0.011 y 9 − 0.007 y10
+0.003 y11 − 0.002 y12 + 0.0007 y13 − 0.0003 y14
+0.11 y 7 − 0.17(− y + 1) y 7 + (0.1(− y + 1)) y8
−(0.07(− y + 1)) y 9 + (0.03(− y + 1)) y10
−(0.01(− y + 1)) y11 + 0.009(− y + 1) y1 2
(35)
−(0.003(− y + 1)) y13 + (0.0005(− y + 1)) y14
(23)
+(1.13(− y + 1)) y 4 − (1.64(− y + 1)) y 5
+(0.7(− y + 1)) y 6 −
H (ν , p) = (1 − p ) [ L(ν ) − L(u0 )] + p [ A(ν ) − f (r )] =
0 (24)
where
(25)
Obviously, considering Eqs. (23) and (24), we have:
H (ν , 0)
 UV ′ − VU ′

 , (29)
p 1
2
 −
′′
(U − Ha U ) 
 Re
−(7.77(− y + 1)) y 2 + (6.54(− y + 1)) y
or
ν (r , p) : Ω× [ 0,1] → R
(28)
+0.00003 y15 − 2.6 y 3 + 3.3 y 2 + 0.22 y 5 − 0.27 y 6
with the boundary condition of:
B (u ,
(27)
and the best approximation is:
p →1
Introducing Eq. (14) in the second momentum equation
(13) shows that quantity ∂Py / ∂y does not depend on the
longitudinal variable x. With the first momentum equation,
we see also that ∂2Py / ∂2x is independent of x. We omit
asterisks for simplicity. Then a separation of variables
leads to [2]
V ′2 − VV ′′ −
ν = ν 0 + p .ν 1 + p 2 .ν 2 + 
L=
(ν=
) − L(u0 ) 0, H (ν ,1) A(ν=
) − f (r ) 0 (26)
where p ∈ [0,1] is an embedding parameter and u0 is the
first approximation that satisfies the boundary condition.
The process of the changes in p from zero to unity is that
of v(r, p) changing from u0 to ur. We consider v as:
d 2U1
+ (1.09(− y + 1)) y 3 =
0,
dy
 −2 y 3   −12 y   −6 y 2   −12 y  d 4V
1 =
−
−12 
−
+
0. (36)

 +3 y 2   +6   +6 y   +6  dy


The solutions of Eqs. (35, 36) may be written as follow:
U1 ( y ) =
−0.000002 y17 + 0.000015 y16 − 0.00006 y15
+0.00013 y14 − 0.0003 y13 + 0.0007 y12 − 0.0014 y11
+0.003 y10 − 0.011 y 9 + 0.04 y8 − 0.06 y 7 + 0.01 y 6
+0.3 y 5 − 0.92 y 4 + 0.9 y 3 + 0.5 y 2 − 0.8 y
(37)
International Journal of Partial Differential Equations and Applications
V1 ( y ) = 0.06 y 7 − 0.2 y 6 − 0.07 y 5
(38)
+ 0.25 y 4 − 5.6 y 3 + 0.25 y 2 .
In the same manner, the rest of components were
obtained by using the Maple package. According to the
HPM, we can conclude:
U ( y) =
−0.000002 y17 + 0.000015 y16 − 0.00006 y15
+0.00013 y14 − 0.0003 y13 + 0.0007 y12 − 0.0014 y11
+0.003 y10 − 0.011 y 9 + 0.04 y8 − 0.06 y 7 + 0.01 y 6
(39)
+0.3 y 5 − 0.92 y 4 + 0.9 y 3 + 0.5 y 2 − 1.8 y + 1
V ( y ) =0.3 y 4 − 0.02 y 8 + 0.011 y 9 − 0.007 y10 + 0.003 y11
−0.001 y12 + 0.0007 y13 − 0.0002 y14 + 0.00004 y15 (40)
−2.6 y 3 + 3.3 y 2 + 0.22 y 5 − 0.27 y 6 + 0.1 y 7
and so on. In the same manner the rest of the components
of the iteration formula can be obtained.
82
4. Results and discussion
In this paper, Similarity Berman’s transformation is
employed to convert the governing partial differential
equations of a steady laminar flow of an electrically
conducting fluid in a two dimensional channel. The HPM
was applied successfully to find the analytical solution of
resulting ordinary differential equation.The numerical
solution is performed using the algebra package Maple
16.0, to solve the present case. The package uses a
Richardson extrapolation procedure for solving nonlinear
boundary value (B-V) problem. Table 1 shows the
comparison between numerical solution and HPM method
for U and V. This is notable that the value of U and V are
obtained by using HPM with 3 times but the values of U
and Vare obtained by using DTM with 20 times [20]. Also
the values obtained by HPM method are closer to the
values obtained by numerically method. So we can
conclude that the HPM method is a powerful analytical
method to solve such this problems.
Table 1. The results of the HPM and numerical methods for V ( y), U(y) for Ha = 1 and Re = 1
y
U ( y)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.0000000000
0.9176926716
0.8384123042
0.7625596488
0.6904130594
0.6221441680
0.5578324646
0.4974789582
0.4410189728
0.3883341514
0.3392636610
0.2936146406
0.2511717980
0.2117062210
0.1749833130
0.1407698920
0.1088404210
0.0789824030
0.0510010310
0.0247230400
0.0000000003
NUM
U ( y)
1.00000000000
0.91816199181
0.83934586197
0.76394633636
0.69223499046
0.62437618918
0.56044187879
0.50042538604
0.44425432048
0.39180262477
0.34290179328
0.29735126495
0.25492797913
0.21539506101
0.17850963207
0.14402974187
0.11172041335
0.08135880743
0.05273855978
0.02567339288
0.00000000000
Figure 2. The comparison between numerical and the HPM solution of
U( y), when Re = 1 and Ha = 1
V ( y)
0.00000000000
0.00785573154
0.03014925679
0.06500661622
0.11060579710
0.16517890600
0.22701243050
0.29444593610
0.36586952870
0.43972039830
0.51447873240
0.58866328860
0.66082687670
0.72955199630
0.79344685820
0.85114197690
0.90128752450
0.94255161390
0.97361964040
0.99319481120
1.0000000000
NUM
V ( y)
0.00000000000
0.00785575099
0.03014932375
0.06500674116
0.11060597119
0.16517910007
0.22701259549
0.29444600642
0.36586942931
0.43972005271
0.51447807609
0.58866228125
0.66082551615
0.72955033156
0.79344499352
0.85114007102
0.90128577938
0.94255024044
0.97361880043
0.99319452777
1.00000000000
Figure 3. The comparison between numerical and the HPM solution of V
( y), when Re = 1 and Ha = 1
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International Journal of Partial Differential Equations and Applications
Figure 4. The comparison between numerical and the HPM solution of V
( y), when Re = 1 and Ha = 0
Figure 5. The comparison between numerical and the HPM solution of
U( y), when Re = 1 and Ha = 0
Figure 6. Effect of Hartman number (Ha) on dimensionless velocities, a) V(y), Re = 1, b) U(y), Re = 1, c) V(y), Re = 5 and d) U(y), Re = 5
International Journal of Partial Differential Equations and Applications
84
Figure 7. Effect of Reynolds number (Re) on dimensionless velocities, a) V(y), Ha = 1, b) U(y), Ha = 1, c) V(y), Ha = 10 and d) U(y), Ha = 10
Figure 2 and Figure 3 show comparison between the
numerical solution and the HPM solution for U( y) and
V( y), when Re =1 and Ha = 1. Figure 4 and Figure 5
illustrate the accuracy of the HPM solution compare to
numerical solution when Re =1 and Ha =0.
Effect of Hartman number (Ha) on dimensionless
velocities is shown in Figure 6. Generally, when the
magnetic field is imposed on the enclosure, the velocity
field suppressed owing to the retarding effect of the
Lorenz force. For example Figure 6(b) shows that as
Hartmann number increases from 0 to 5, velocity
boundary layer thickness decreases. For low Reynolds
number, as Hartmann number increases V(y) decreases for
y > ym but opposite trend is observed for y < ym, ym is a
meeting point that all curves joint together at this point.
When Reynolds number increases this meeting point shifts
to the solid wall and it can be seen that V(y) decreases
with increase of Hartmann number.
Effect of Reynolds number (Re) on dimensionless
velocities is shown in Figure 7. As seen in Figure 7, when
Ha number is small the effect of Re number is more
sensible and increasing the Re number, makes a decrease
in velocity profiles also it is worth to mention that the
Reynolds number indicates the relative significance of the
inertia effect compared to the viscous effect. Thus,
velocity boundary layer thickness on lower plate decreases
as Re increases and in turn increasing Re leads to an
increase in the magnitude of the skin friction coefficient.
5. Conclusion
In this paper, Homotopy perturbation method is used to
solve the problem of laminar fluid flow in a semi-porous
channel in the presence of uniform magnetic field. In this
case study, the Berman’s similarity transformation has
been used to reduce the governing differential equations
into a set of coupled ordinary non-linear differential
equations. In most cases, these problems do not admit
analytical solution, so these equations should be solved
using special techniques.It has been attempted to exhibit
the reliability and performance of the Homotopy
perturbation method (HPM) in comparison with the
numerical method (Richardson extrapolation) in solving
this problem. The results indicate that velocity boundary
layer thickness decrease with increase of Reynolds
number and it increases as Hartmann number increases.
85
International Journal of Partial Differential Equations and Applications
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